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In mathematics, two-center bipolar coordinates is a coordinate system, based on two coordinates which give distances from two fixed centers, $$c_1$$ and $$c_2 \(. This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane). Transformation to Cartesian coordinates Cartesian coordinates and polar coordinates. The transformation to Cartesian coordinates \( (x,\ y)$$ from two-center bipolar coordinates $$(r_1,\ r_2)$$ is

$$x = \frac{r_2^2-r_1^2}{4a}$$

$$y = \pm \frac{1}{4a}\sqrt{16a^2r_2^2-(r_2^2-r_1^2+4a^2)^2}$$

where the centers of this coordinate system are at $$(+a,\ 0)$$ and $$(-a,\ 0)$$.
Transformation to polar coordinates

When x>0 the transformation to polar coordinates from two-center bipolar coordinates is

$$r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}$$

$$\theta = \arctan \left(\frac{\sqrt{r_1^4-8a^2r_1^2-2r_1^2r_2^2-(4a^2-r_2^2)^2}}{r_2^2-r_1^2} \right)\,\!$$

where 2 a is the distance between the poles (coordinate system centers).

Biangular coordinates
Lemniscate of Bernoulli
Oval of Cassini
Cartesian oval
Ellipse

References

Weisstein, Eric W., "Bipolar coordinates", MathWorld.
R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.
The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.